Lecture 2: Signals Concepts & Properties (1) Systems, signals, mathematical models. Continuous-time and discrete-time signals. Energy and power signals. Linear systems. Examples for use throughout the course, introduction to Matlab and Simulink tools Specific objectives for this lecture include General properties of signals Energy and power for continuous & discrete-time signals Signal transformations Specific signal types Representing signals in Matlab and Simulink ELEC 405, L2

Reminder: Continuous & Discrete Signals Continuous-Time Signals Most signals in the real world are continuous time, as the scale is infinitesimally fine. E.g. voltage, velocity, Denote by x(t), where the time interval may be bounded (finite) or infinite Discrete-Time Signals Some real world and many digital signals are discrete time, as they are sampled E.g. pixels, daily stock price (anything that a digital computer processes) Denote by x[n], where n is an integer value that varies discretely Sampled continuous signal x[n] =x(nk) x(t) t x[n] n ELEC 405, L2

“Electrical” Signal Energy & Power It is often useful to characterise signals by measures such as energy and power For example, the instantaneous power of a resistor is: and the total energy expanded over the interval [t1, t2] is: and the average energy is: How are these concepts defined for any continuous or discrete time signal? ELEC 405, L2

Generic Signal Energy and Power Total energy of a continuous signal x(t) over [t1, t2] is: where |.| denote the magnitude of the (complex) number. Similarly for a discrete time signal x[n] over [n1, n2]: By dividing the quantities by (t2-t1) and (n2-n1+1), respectively, gives the average power, P Note that these are similar to the electrical analogies (voltage), but they are different, both value and dimension. ELEC 405, L2

Energy and Power over Infinite Time For many signals, we’re interested in examining the power and energy over an infinite time interval (-∞, ∞). These quantities are therefore defined by: If the sums or integrals do not converge, the energy of such a signal is infinite Two important (sub)classes of signals Finite total energy (and therefore zero average power) Finite average power (and therefore infinite total energy) Signal analysis over infinite time, all depends on the “tails” (limiting behaviour) ELEC 405, L2

Time Shift Signal Transformations A central concept in signal analysis is the transformation of one signal into another signal. Of particular interest are simple transformations that involve a transformation of the time axis only. A linear time shift signal transformation is given by: where b represents a signal offset from 0, and the a parameter represents a signal stretching if |a|>1, compression if 0<|a|<1 and a reflection if a<0. ELEC 405, L2

Periodic Signals An important class of signals is the class of periodic signals. A periodic signal is a continuous time signal x(t), that has the property where T>0, for all t. Examples: cos(t+2p) = cos(t) sin(t+2p) = sin(t) Are both periodic with period 2p NB for a signal to be periodic, the relationship must hold for all t. 2p ELEC 405, L2

Odd and Even Signals An even signal is identical to its time reversed signal, i.e. it can be reflected in the origin and is equal to the original: Examples: x(t) = cos(t) x(t) = c An odd signal is identical to its negated, time reversed signal, i.e. it is equal to the negative reflected signal x(t) = sin(t) x(t) = t This is important because any signal can be expressed as the sum of an odd signal and an even signal. ELEC 405, L2

Exponential and Sinusoidal Signals Exponential and sinusoidal signals are characteristic of real-world signals and also from a basis (a building block) for other signals. A generic complex exponential signal is of the form: where C and a are, in general, complex numbers. Lets investigate some special cases of this signal Real exponential signals Exponential growth Exponential decay ELEC 405, L2

Periodic Complex Exponential & Sinusoidal Signals Consider when a is purely imaginary: By Euler’s relationship, this can be expressed as: This is a periodic signals because: when T=2p/w0 A closely related signal is the sinusoidal signal: We can always use: cos(1) T0 = 2p/w0 = p T0 is the fundamental time period w0 is the fundamental frequency ELEC 405, L2

Exponential & Sinusoidal Signal Properties Periodic signals, in particular complex periodic and sinusoidal signals, have infinite total energy but finite average power. Consider energy over one period: Therefore: Average power: Useful to consider harmonic signals Terminology is consistent with its use in music, where each frequency is an integer multiple of a fundamental frequency ELEC 405, L2

General Complex Exponential Signals So far, considered the real and periodic complex exponential Now consider when C can be complex. Let us express C is polar form and a in rectangular form: So Using Euler’s relation These are damped sinusoids ELEC 405, L2

Discrete Unit Impulse and Step Signals The discrete unit impulse signal is defined: Useful as a basis for analyzing other signals The discrete unit step signal is defined: Note that the unit impulse is the first difference (derivative) of the step signal Similarly, the unit step is the running sum (integral) of the unit impulse. ELEC 405, L2

Continuous Unit Impulse and Step Signals The continuous unit impulse signal is defined: Note that it is discontinuous at t=0 The arrow is used to denote area, rather than actual value Again, useful for an infinite basis The continuous unit step signal is defined: ELEC 405, L2

Lecture 2: Summary This lecture has looked at signals: Power and energy Signal transformations Time shift Periodic Even and odd signals Exponential and sinusoidal signals Unit impulse and step functions Matlab and Simulink are complementary environments for producing and analysing continuous and discrete signals. This will require some effort to learn the programming syntax and style! ELEC 405, L2